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<h4 class="heading"><span class="type">Paragraph</span></h4>
<p>One can see that</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/Eq6_1.html ./knowl/Eq7_1.html">
\begin{equation}
\lim_{\tau \to 0} d_{\tau}(t)=0,\quad t\neq 0.\tag{8.5.1}
\end{equation}
</div>
<p class="continuation">And let</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/Eq6_1.html ./knowl/Eq7_1.html">
\begin{equation*}
I(\tau)=\int_{-\infty}^{\infty} d_{\tau}(t) \mathrm{d}t,
\end{equation*}
</div>
<p class="continuation">then it is to calculate <span class="process-math">\(I(\tau)=1\)</span> for each <span class="process-math">\(\tau \neq 0\)</span> and</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/Eq6_1.html ./knowl/Eq7_1.html">
\begin{equation}
\lim_{\tau \to 0} I(\tau)=1.\tag{8.5.2}
\end{equation}
</div>
<p class="continuation">Equations (<a href="" class="xref" data-knowl="./knowl/Eq6_1.html" title="Equation 8.5.1">(8.5.1)</a>) and (<a href="" class="xref" data-knowl="./knowl/Eq7_1.html" title="Equation 8.5.2">(8.5.2)</a>) can be used to define the unit impulse function <span class="process-math">\(\delta\)</span> which have the properties</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/Eq6_1.html ./knowl/Eq7_1.html">
\begin{equation*}
\delta(t)=0,\quad t \neq 0
\end{equation*}
</div>
<p class="continuation">and</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/Eq6_1.html ./knowl/Eq7_1.html">
\begin{equation*}
\int_{-\infty}^{\infty} \delta(t) \mathrm{d} t=1.
\end{equation*}
</div>
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